16. At a California border inspection station, vehicles arrive at the rate of2 p
ID: 344691 • Letter: 1
Question
16. At a California border inspection station, vehicles arrive at the rate of2 per hour in a Poisson distribution. For simplicity in this problem, assume that there is only one lane and one inspector, who can inspect vehicles with average exponentially distributed time of 15 minutes. (12 points) a. What is the probability that the inspector will be idle? b. What is the average no. of vehicles waiting for inspection? c. What is the average time a vehicle spends in the system?? d. What is the probability that there are three or more vehicles in the system?Explanation / Answer
Let ,
Inspection rate ( @ 15 minutes per inspection ) = S = 4 / hour
Arrival rate of vehicles =a = 2 / hour
= Probability that there will be no vehicles
= Po
= 1 – a/s
= 1 – 2/4
= 0.5
PROBABILITY THAT INSPECTOR WILL BE IDLE = 0.5
= a^2/ s x ( s – a)
= ( 2 x 2 ) / 4 x ( 4 – 2 )
= 4/8
= 0.5
AVERAGE NUMBER OF VEHICLS WAITING FOR INSPECTION = 0.5
= a/s x ( s – a) + 1/shour
= 2/( 4 x 2 ) + 1 / 4 hour
= 1 / 4 + 1 / 4 hour
= ½ hour
= 30 minutes
AVERAGE TIME A VEHICL SPENDS IN THE SYSTEM = 30 MINUTES
Probability that there will be 2 vehicles in the system = P2 = ( a/s)^2 x P0 = 0.5 x 0.5 x 0.5 = 0.125
Probability that there will be maximum 2 vehicles in the system
= Po + P1 + P2
= 0.5 + 0.25 + 0.125
= 0.875
Probability that there will be 3 or more vehicles in the system
= 1 – Probability that there will be maximum 2 vehicles in the system
= 1 – 0.875
= 0.125
PROBABILITY THAT THERE WILL BE 3 OR MORE VEHICLES IN THE SYSTEM = 0.125
PROBABILITY THAT INSPECTOR WILL BE IDLE = 0.5
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